# Pre-registration — the kill shadow's drift AMPLITUDE **Written 2026-08-19, TODO item 5's one remaining open number, BEFORE any new measurement was run.** Committed alone, before either producer existed (`research/shadow-amplitude-01-derivation.js`, `research/shadow-amplitude-02-measure.js`). ## Custody, stated plainly This is not a blind pre-registration and must not be read as one. The ten measured cluster depths and their slot counts were **already embedded** in `research/shadow-buchstab-02-instrument.js` and re-verified in `research/adversary-wave2-01-shadow.js` before this file was written, and they were read. Under the standing compute rule they are cited here, not re-measured. What is therefore fixed in advance and *cannot* be tuned: 1. the **form** of the correction — it has no free parameter, it is a product over primes, and it is written out in full below; 2. the **corrected prediction per cluster**, frozen as a table of ten numbers; 3. the **scoring rules**, including the error model and the tolerances; 4. two quantities that have **never been measured anywhere in this corpus** and are genuinely blind here: the pooled pre-band depth `[y²/2, y²)` over a whole cluster, and the exact number of *distinct* pairs a cluster's overlapping bands actually sample. The frozen numbers in §2 and §3 were computed in a scratchpad script before this file was committed; `shadow-amplitude-01-derivation.js` reproduces them digit for digit from committed code, and no measured band datum enters that producer. ## 0. The object `adversary-wave2.md` §1 correction 2: the shadow's depth SHAPE is right (drift coefficient `2 − 1/ln 2`, D3 10 of 10) but the drift's AMPLITUDE is not. Measured drift `−0.03207` against a predicted `−0.01211` on the independent ladder binning (ratio **2.648**), `−0.02726` against `−0.01188` on the record's own clusters (**2.295**), `1.186` restricted to `y ≥ 2000`. `shadow-buchstab.md` §4's candidate explanation is that "the local pair density is still approaching its Hardy–Littlewood form at `x ~ 10⁶`", with the note that the residual "decays too fast for a `1/ln y` or `1/ln²y` term". ## 1. The hypothesis, derived The instrument normalises every count by the **exact** tile density delta(y) = (1/2) prod_{2y}(1 - 1/(p-1)^2)`, `C2 = prod_{p>2}(1 - 1/(p-1)^2)`. From `(1 - 2/p) = (1 - 1/(p-1)^2)(1 - 1/p)^2`, delta(y) = 2 C2 M(y)^2 / A(y), delta_inf(y) := 2 C2 e^{-2 gamma} / (ln y)^2, K(y) := delta(y)/delta_inf(y) = (e^gamma M(y) ln y)^2 / A(y). `K` is exactly the second-order content of the pair-Mertens product over the prime range the band sieves: the squared Mertens partial-product error, divided by the twin-constant tail. The normalisation `delta_inf . rho` is the exact Hardy–Littlewood local pair density — at `u = 2`, `delta_inf . e^{2 gamma}/4 = 2 C2 / (2 ln y)^2 = 2 C2 / (ln x)^2`. So measured(y) = lambda(y) . B_x(y) / K(y), where `lambda(y)` is the ratio of the true local pair density in the band to its Hardy–Littlewood form. **The corrected prediction is `B_x(y)/K(y)`, and `lambda` is the only remaining freedom.** `K < 1` at every level reached, so the correction is positive, which is the sign of the observed residual. Two corollaries that the correction must not disturb: `1/K` is a level-wide factor, so it cannot touch D3 (which passes 10 of 10) and it cannot touch the coefficient `2 - 1/ln 2` (a first moment inside the band). ## 2. The corrected prediction, frozen Clusters, level lists and `B_x` are those of `shadow-buchstab-02-instrument.js` part (B) (centres `y0`, pool `[0.88 y0, 1.12 y0]`, budget `4e8`). `measured` is that file's embedded column, cited. | y0 | levels | range | B_x | mean 1/K | **B_corr** | measured | measured − B_corr | |---|---|---|---|---|---|---|---| | 1000 | 36 | 881–1117 | 0.83365 | 1.00967 | **0.84171** | 0.85004 | +0.00833 | | 1400 | 45 | 1237–1567 | 0.83192 | 1.00728 | **0.83797** | 0.84124 | +0.00327 | | 2000 | 59 | 1777–2239 | 0.83025 | 1.00584 | **0.83510** | 0.83285 | −0.00225 | | 2900 | 53 | 2557–2963 | 0.82886 | 1.00520 | **0.83317** | 0.83212 | −0.00105 | | 4200 | 28 | 3697–3919 | 0.82756 | 1.00429 | **0.83111** | 0.82830 | −0.00281 | | 6000 | 14 | 5281–5413 | 0.82629 | 1.00218 | **0.82809** | 0.82690 | −0.00119 | | 8500 | 8 | 7481–7529 | 0.82512 | 1.00163 | **0.82647** | 0.82741 | +0.00094 | | 12000 | 4 | 10567–10601 | 0.82401 | 1.00223 | **0.82585** | 0.82325 | −0.00260 | | 18000 | 2 | 15859–15877 | 0.82279 | 1.00207 | **0.82449** | 0.82315 | −0.00134 | | 26000 | 1 | 22901 | 0.82177 | 1.00138 | **0.82290** | 0.82279 | −0.00011 | Ladder binning (`adversary-wave2-01-shadow.js` part C, one prime per rung): `1/K = 1.00704` at `y = 1009` and `1.00111` at `y = 25409`, so `B_corr` runs `0.83947 → 0.82241` and the corrected ladder drift is **−0.01707** against a measured `−0.03207`. ## 3. The error model, fixed in advance The record's `se(Poisson) = measured/sqrt(Ntot)` is computed on the **pooled** slot total, and the pooling oversamples: at `y0 = 1000` thirty-six bands `[y², 2y²]` covering `[776161, 2495378]` are summed, so the same pair is counted up to eighteen times and `Ntot = 266735` counts about 13,000 distinct pairs. The counting floor used for scoring is therefore sigma_eff = measured / sqrt(N_dist), with `N_dist` the **exact** count of pairs the cluster's bands sample at least once. A pair `n` is sampled iff `n` is `y_lo`-rough where `y_lo` is the smallest cluster level with `n < 2 y_lo^2` (roughness is decreasing in `y`, so the smallest covering level is the weakest test), which makes `N_dist` one segmented pass over the union interval. Provisional estimates `N_dist ~ delta(y_min) . |union| . measured` give sigma_eff ~ 0.00743, 0.00553, 0.00405, 0.00335, 0.00281, 0.00218, 0.00166, 0.00123, 0.00086, 0.00062 for the ten clusters; the exact counts replace them at scoring. Poisson is declared as a **floor**, not an estimate: prime pairs in long intervals are not Poisson, and the correlation between overlapping bands can only widen the bar. ## 4. Criteria **C1 — amplitude (the object).** `R = (measured drift)/(B_corr drift)` over the full cluster span. PASS iff `1` lies inside the 2-sigma interval of `R` propagated from the two endpoint clusters' `sigma_eff`. Reported for the full span, for `y >= 2000`, and for the ladder. Frozen: the same test on the **uncorrected** prediction gives `R = 2.294` with 2-sigma interval `[1.039, 3.549]`, which excludes 1; on the corrected prediction `R = 1.449` with `[0.656, 2.242]`. **C2 — per-cluster residuals.** PASS iff `|measured − B_corr| <= 2 sigma_eff` at all ten clusters AND the mean of `(measured − B_corr)/sigma_eff` lies in `[−1, +1]`. **C3 — route (a) meets route (b), and this one is blind.** The pre-band `[y²/2, y²)` is pooled over every level of every cluster. Below `y²` a level-`y` slot is a twin prime and nothing else, so the pre-band's model is the `u <= 2` branch `e^{2 gamma}/u²` — exact Hardy–Littlewood, no independence conjecture in it. Then `lambda_pre := measured_pre / pred_pre = lambda_twin / K`. Prediction: `lambda_pre` equals the frozen `mean 1/K` column of §2 at every cluster. PASS iff that holds within `2 sigma_eff(pre-band)` at 8 or more of 10 clusters. **C4 — adjudicating the record's own explanation, blind.** Set `lambda_twin = lambda_pre . K`, the measured local twin density over its Hardy–Littlewood form at each cluster's scale. - *"still approaching HL"* (the record's reading) predicts `lambda_twin > 1` at every cluster with a monotone decay toward 1. - *"oscillates about HL"* predicts mixed signs and a mean within `2 sigma` of 1. PASS for the correction hypothesis iff the second holds; the first would mean a second systematic term survives and its size and decay must be stated. **C5 — kill.** The hypothesis is WRONG if C1 fails, or if `measured − B_corr` carries a trend in `y` significant at 2 sigma, or if `lambda_pre` differs from `1/K` in the same direction at 8 or more clusters beyond `2 sigma`. ## 5. Verdicts - **AMPLITUDE DERIVED** — C1, C2, C3 and C4 all pass. The drift's amplitude is the pair-Mertens finite-`y` factor `1/K(y)`; the shadow closes entirely, and the residual is counting noise whose true scale the record understated. - **PARTIALLY EXPLAINED** — C1 passes, one of C2/C3/C4 fails. State the remaining term's size at `y ~ 1000` and its decay, and say what it correlates with. - **HYPOTHESIS WRONG** — C1 fails or C5 triggers. The residual is structure; name what it correlates with. No statistic outside C1–C5 will be promoted to a verdict after the fact. The `[2,3]` branch of the survival curve remains the independence-squared conjecture of `attack2-05` throughout; nothing scored here is a theorem about it.